Influence of initial states on memory effects: A study of early-time superradiance

The initial state of a quantum system can significantly influence its future dynamics [1, 2], especially in non-Markovian quantum processes due to the environmental memory effects. One trivial example is that if the system is initially in a steady state in a non-Markovian quantum process, it can hardly exhibit any non-Markovian features afterward, such as the nonmonotonic behaviors of energy and information flows [3, 4, 5, 6]. More intriguing phenomena may emerge when a system consists of a collection of subsystems, such that the properties of its initial state, such as entanglement and coherence, may significantly influence its future dynamics. A well-known example is the concept of superradiance [7, 9, 8, 10, 11, 12, 13, 14] introduced by Dicke in 1954, where the emission intensity from an ensemble of atoms interacting with a common electromagnetic field can be enhanced compared with that from independent atoms. It is also well-known that the superradiance behaviors are highly relevant to the initial states of the system. In recent years, superradiance has received a large amount of attention due to its theoretical significance and potential applications [16, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Under certain approximations, such as a coarse-grained timescale, the superradiance process could be regarded as Markovian [10, 11, 12, 13, 14], whereas, it is intrinsically non-Markovian. With the advances in theories and technologies, understanding the non-Markovian dynamics of superradiance becomes more demanding [28, 29, 30, 31]. In a recent work [31], the author shows that non-Markovian memory effects play an important role in superradiance beyond retardation, featuring the quadratic dynamics in the early-time (Zeno) regime, whereas, the memory effects are not quantitatively evaluated. In view of the significant influences of the system initial states on non-Markovian quantum processes, especially, a superradiance process, some interesting questions arise. For example, how to quantitatively evaluate the memory effects of a quantum process conditioned on a particular system initial state? Are there quantitative relations between the memory effects and the superradiance characteristics in the early-time regime? What is the role of the (very weak) memory effects in a superradiance process that could be well approximated as Markovian?

In recent years, a number of measures or manifestations of non-Markovianity were proposed to characterize the memory effects which are often connected to nonmonotonic behaviors [3, 4, 5, 6]. For example, the well-known BLP measure [32] and the RHP measure [33] use the increases of distinguishability and entanglement, respectively, to measure the non-Markovianity. The nonmonotonic behaviors originate from the noncomplete positivity of the intermediate dynamical maps in non-Markovian processes. These measures generally apply to quantum processes where the evolutions start at a fixed time $t_{0}$ (usually $t_{0}=0$ for simplicity). However, a key distinction of the non-Markovian process is the existence of $t_{0}$ in the generator of the system’s time-local equation, i.e., $\dot{\rho}_{S}=\mathcal{L}(t-t_{0})\rho_{S}$ [34]. Thus some features of non-Markovianity might not be characterized with only evolutions starting from a fixed time $t_{0}$. For example, in the early-time regime or a near-Markovian regime, the spontaneous radiation of an atom could be described by a time-local equation with a positive (time-dependent) damping rate, which looks like a time-dependent Markovian master equation (except $t_{0}$ in its generator) [35]. Thus the nonmonotonic behaviors do not occur in these regimes, implying weaker memory effects. Meanwhile, the non-Markovianity measures based on the noncomplete positivity may give zero memory effects. On the other hand, the previously proposed non-Markovianity measures generally do not depends on a particular system initial state, but deal with a quantum process where all the potential initial states are considered. Therefore, an initial-state-dependent measure which is able to characterize weak memory effects is demanding.

In this paper, we propose such a method developed from a previous work [35] of ours. The non-Markovianity measure in Ref. [35] is based on the inequality of completely positive dynamical maps $T(t_{2},t_{0})\neq T(t_{2},t_{1})T(t_{1},t_{0})$ and connected with the validity of the Markov approximation. By acting the inequality on a particular initial state $\rho_{S}(t_{0})$, the evolutions starting at $t_{0}$ and $t_{1}$ have the same system states at $t_{1}$ but different histories before $t_{1}$. Thus the memory effects (past-future dependence) could be understood by the difference between the final states $\rho_{S}(t_{2})$ and $\rho_{S}^{\prime}(t_{2})$. Based on the above interpretation, we suggest measuring the memory effects conditioned on the initial state $\rho_{S}(t_{0})$ by the maximal difference of $\rho_{S}(t_{2})$ and $\rho_{S}^{\prime}(t_{2})$ in a time interval of interest. Using this method, we calculate the memory effects (as well as the radiation characteristics) of a superradiance model with two types of initial states: the Dicke states and factorized identical states in different regimes. The model describes $N$ two-level atoms interacting with a single-mode cavity initially in vacuum. We give analytic expressions of the characteristics of memory effects (and superradiance) in the early-time regime. Moreover, we extend our work to longer time intervals in the near-Markovian regime, and to the whole radiation lifetime in a strongly non-Markovian regime with numerical results.

The main findings are as follows. 1. The radiation intensity is closely related to the memory effects, and correspondingly, the degree of superradiance is closely related to the degree of memory-effect-enhancement (from independent radiation to collective radiation), especially for the Dicke states. 2. The entanglement in the Dicke states and the single-atom coherence in the factorized identical states are necessary for the superradiance and important for the memory-effect-enhancement. 3. The (change of) environmental photon number is a main physical source of memory effects for our model. 4. The characteristics of memory effects and superradiance in long-time dynamics (especially for the near-Markovian regime) can be reflected through its early-time dynamics.

The paper is organized as follows: In Sec. II, we propose our method to evaluate the initial-state-dependent memory effects in a quantum process. In Sec. III, we obtain the early-time solution of $N$ two-level atoms interacting with a vacuum cavity, with which we give analytic expressions for the value of memory effects, the cavity photon number, the degree of superradiance, and so on. In Sec. IV, the influences of two types of initial states on the memory effects as well as the superradiance are calculated and analyzed using the early-time results. Our work is extended to the near-Markovian regime and a strongly non-Markovian regime in Sec. V. At last, conclusions and discussions are presented in Sec. VI.

In this section, we first review the measure of non-Markovianity in Ref. [35], which quantifies the memory effects (the past-future dependence) in a quantum process. Using its physical interpretations, we then propose a method to quantify the memory effects in a quantum process conditioned on a particular initial state of the system. The object of study in Ref. [35] is a quantum process of an open quantum system described by the total Hamiltonian

an arbitrary system initial state $\rho_{S}(t_{I})$ and a fixed initial state of the environment $\rho_{E}(t_{I})$ (independent of the system). Here $t_{I}$ is the initial time of an evolution which is also arbitrary. The initial condition of an evolution is

which means that the system is initially uncorrelated with the environment before the evolution. In general, $\rho_{E}(t_{I})$ is governed by [6]

Typically, one deals with a time-independent environment Hamiltonian $H_{E}$ and assumes a steady state of $H_{E}$ as the environmental initial state (e.g., a thermal state). Then, the initial condition of an evolution is

for any $t_{I}$. Since $\rho_{S}(t_{I})$ and $t_{I}$ is arbitrary, the measure of non-Markovianity (memory effects) in Ref. [35] is determined by $H$ and $\rho_{E}(t_{I})$ and the time interval of interest.

In a non-Markovian quantum process, the meaning of a dynamical map given by $\rho_{S}(t_{2})=\varepsilon(t_{2},t_{1})\rho_{S}(t_{1})$ is ambiguous unless $\rho_{SE}(t_{1})$, or from another perspective, the initial time $t_{I}$ is specified ($t_{I}\leqslant t_{1}$). For clarity, we define $T(t_{2},t_{1})$ as a memoryless dynamical map that transfers $\rho_{S}(t_{1})$ to $\rho_{S}(t_{2})$, where $t_{1}$ is the initial time of the evolution ($t_{I}=t_{1}$) [35], i.e.,

where $\rho_{E}(t_{1})$ is fixed, $\rho_{S}(t_{1})$ is arbitrary and $U(t_{2},t_{1})=\mathcal{T}e^{-i\int_{t_{1}}^{t_{2}}H(\tau)d\tau}$ in general. Particularly, when $H$ is time-independent and $\rho_{E}(t_{I})=\rho_{E}$ is a steady state of $H_{E}$, Eq. (5) is simplified to

such that $T(t_{2},t_{1})=T(t_{2}-t_{1},0)$ [34]. The dynamical map $T$ is trace-preserving and completely positive and called a universal dynamical map (UDM) that is independent of the state it acts upon [36].

Let $t_{0}\leqslant t_{1}\leqslant t_{2}$, a (time-dependent) quantum process is Markovian if the dynamical maps (denoted by $\Lambda_{M}$) satisfy the divisibility condition

where each dynamical map is uniquely defined and a UDM. Remark that the dynamical map $\Lambda_{M}(t_{2},t_{1})$ is a UDM if and only if it is induced by

where $\rho_{E}(t_{1})$ is fixed and $\rho_{S}(t_{1})$ is arbitrary [36]. In an open quantum system, the condition $\rho_{SE}(t)=\rho_{S}(t)\otimes\rho_{E}(t)$ [$\rho_{E}(t)$ does not dependent on the system state] may not be satisfied exactly for $t>t_{I}$. Thus the dynamics of an exact open quantum system is typically not Markovian [36]. However, $\rho_{SE}(t)\approx\rho_{S}(t)\otimes\rho_{E}(t)$ can be a good approximation where the correlation between the system and the environment does not affect the system’s dynamics so much [36]. It is observed that the Markovian dynamical map $\Lambda_{M}(t_{2},t_{1})$ is unique and does not depend on the initial time of an evolution, e.g., $t_{I}=t_{1}$ or $t_{I}<t_{1}$. Therefore, $\Lambda_{M}(t_{2},t_{1})=T(t_{2},t_{1})$ according to the definition of the memoryless dynamical map $T$. Then, the Markovian divisibility condition (7) can also be expressed in terms of the memoryless dynamical map $T$ by

whose violation is a sign of non-Markovianity. Unlike the dynamical maps used in some non-Markovianity measures [33, 37], all the dynamical maps $T$ are completely positive. The violation of Eq. (9) is manifested by the inequality

The physical meaning of Eq. (10) can be explained with Fig. 1. Let the left-hand and the right-hand sides of Eq. (10) act on a system initial state $\rho_{S}(t_{0})$. On the left-hand side of Eq. (10), $\rho_{S}(t_{0})$ is mapped to $\rho_{S}(t_{2})$ by $T(t_{2},t_{0})$ in evolution A. On the right-hand side, $\rho_{S}(t_{0})$ is first mapped to $\rho_{S}(t_{1})$ by $T(t_{1},t_{0})$ in evolution B. Then, as the initial state of evolution C, $\rho_{S}(t_{1})$ is mapped to $\rho^{\prime}_{S}(t_{2})$ by $T(t_{2},t_{1})$, which means that the initial condition of evolution C is $\rho_{SE}(t_{1})=\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})$ with fixed $\rho_{E}(t_{1})$ defined by Eq. (3) or a steady one $\rho_{E}$. At moment $t_{1}$, evolution A and C have the same system state $\rho_{S}(t_{1})$ but different histories: evolution A has a history in time interval $[t_{0},t_{1}]$, which is encoded in $\rho_{SE}(t_{1})=U(t_{1},t_{0})\rho_{S}(t_{0})\otimes\rho_{E}(t_{0})U(t_{1},t_{0})^{{\dagger}}$, while evolution C (starting at $t_{1}$) has no history before $t_{1}$. Therefore, $\rho_{S}(t_{2})\neq\rho^{\prime}_{S}(t_{2})$ is an evidence that the future state (after $t_{1}$) of the system depends on its history (in $[t_{0},t_{1}]$) in this quantum process. This is the physical meaning of the memory effects in this paper and [35]. Otherwise, if the divisibility (9) holds, the process is Markovian and $\rho_{S}(t_{2})=\rho^{\prime}_{S}(t_{2})$ for any $\rho_{S}(t_{0})$.

On the other hand, the inequality could be understood by focusing on the change of the environment state between evolution B and C. At the end of evolution B, $\rho_{SE}(t_{1})=U(t_{1},t_{0})\rho_{S}(t_{0})\otimes\rho_{E}(t_{0})U(t_{1},t_{0})^{{\dagger}}$. Then, at the beginning of evolution C, the environment is initialized by $T(t_{2},t_{1})$ such that $\rho_{SE}(t_{1})\rightarrow\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})$, where $\rho_{E}(t_{1})$ is independent of the system. The information of the system’s history in $[t_{0},t_{1}]$ is erased by $T(t_{2},t_{1})$. In contrast, such an initialization never happens in evolution A. Therefore, $\rho_{S}(t_{2})\neq\rho^{\prime}_{S}(t_{2})$ signifies that the environment (as well as the correlations between the system and environment [36]) remembers the system’s history and this memory can influence the future of the system.

In Ref. [35], we define the maximal difference of $T(t_{2},t_{0})$ and $T(t_{2},t_{1})T(t_{1},t_{0})$ as the non-Markovianity where $t_{1}$ and $t_{2}$ are optimized. Based on the dynamical maps $T$, all the initial states are potentially considered, thus the measure of non-Markovianity in Ref. [35] does not depend on the initial state of the system. In this paper, our goal is to evaluate the influence of the system initial state on the memory effects in a quantum process. Using the physical interpretations discussed above, we define the value of memory effects conditioned on $\rho_{S}(t_{0})$ as the maximal trace distance between $\rho_{S}(t_{2})$ and $\rho^{\prime}_{S}(t_{2})$:

Here $||A||=\text{Tr}(\sqrt{A^{\dagger}A})$ is the trace norm of an operator $A$. Assuming a fixed $t_{0}$ for convenience, $N_{M}[\rho_{S}(t_{0})]$ could be calculated by optimizing $t_{1}$ and $t_{2}$ in a time interval $[t_{0},t]$ of interest where $t_{0}\leqslant t_{1}\leqslant t_{2}\leqslant t$. For example, the time interval might be $[t_{0},\infty]$ or $[t_{0},t_{0}+\tau]$, where $\tau$ is finite (as done later in this paper). Similar to the measure in Ref. [35], $0\leqslant N_{M}[\rho_{S}(t_{0})]\leqslant 1$ is satisfied due to the properties of the trace distance.

Using Eq. (11), the influence of initial states on the memory effects of a quantum process could be quantitatively evaluated. Note that $N_{M}[\rho_{S}(t_{0})]>0$ is a sufficient condition for the inequality Eq. (10), but not a necessary one. The theoretical calculation and experimental observation of Eq. (11) might be easier than those in Ref. [35] since the determination of the dynamical maps $T$ is not compulsory. Moreover, as discussed in Ref. [35], $\text{Tr}[P\rho(t_{2})]\neq\text{Tr}[P\rho^{\prime}(t_{2})]$ is a sufficient condition for $\rho_{S}(t_{2})\neq\rho_{S}^{\prime}(t_{2})$, where $P$ is an operator of a physical quantity. Therefore, when the full information of the system state is not easily obtainable, $\Delta P[\rho_{S}(t_{0})]=|\text{Tr}[P\rho(t_{2})]-\text{Tr}[P\rho^{\prime}(t_{2})]|$ ($\max_{t_{1},t_{2}}\Delta P[\rho_{S}(t_{0})]$) could be used as an evidence (manifestation) of the memory effects for simplicity and intuitiveness both theoretically and experimentally.